5
$\begingroup$

Let $ (X,\mathcal{F},\mu) $ and $ (G,\mathcal{G},\nu) $ be two measure spaces with $ \mu $ and $ \nu $ being $ \sigma $-finite. Per definition, the linear span of $$ \{ \mathbf{1}_{C} ~|~ C \in \mathcal{F} \otimes \mathcal{G} ~ \text{and} ~ (\mu \otimes \nu)(C) < \infty \} $$ is dense in $ {L^{p}}(X \times Y,\mathcal{F} \otimes \mathcal{G},\mu \otimes \nu) $ for any $ p \in [1,\infty) $. This should also be true for the linear span of $$ \{ \mathbf{1}_{A \times B} ~|~ A \in \mathcal{F}, ~ B \in \mathcal{G} ~ \text{and} ~ \mu(A),\nu(B) < \infty \}. $$ Do you know a reference?

Thank you.

Improve this question
$\endgroup$
1
  • $\begingroup$ Real Analysis: Modern techniques and their applications by G.B.Folland is a good reference. $\endgroup$
    – Uday
    Sep 3, 2012 at 11:51

2 Answers 2

Reset to default
6
$\begingroup$

A useful density lemma is the following.

Let $(X,\mathcal{A}, \mu)$ be a measure space and let $\Gamma\subset\mathcal{P}(X)$ a ring of sets of finite measure that generates the $\sigma$-algebra $\mathcal{A}$. Then, the linear span of the characteristic functions of sets in $\Gamma$ is dense in $L^p$ (here $1\le p < +\infty$)

In your case, of course, you can take $\Gamma$ to be the collection of finite unions of Cartesian products of measurable sets of finite measure.

Incidentally, the hypothesis on $\Gamma$ in that density lemma can be weakened (one does not need all the ring strucure of $\Gamma$). Let's say that $\Gamma\subset \mathcal{P}(X)$ is a "semi-ring" (warning: not standard; I borrowed it from Halmos, with a slightly more general meaning) if the following holds:

For all $A$ and $B$ in $\Gamma$ the sets $A\setminus B$ and $A\cap B$ are both expressible as union of countably many disjoint element of $\Gamma$.

Then the above lemma holds true. The notion of "semi-ring"is also interesting, in that it is a convenient domain for a completely additive set function, in order that the Caratheodory's Extension Theorem holds.

Improve this answer
$\endgroup$
5
  • $\begingroup$ Thank you, do you have a reference for the first Lemma? Best, warsaga $\endgroup$
    – warsaga
    Sep 3, 2012 at 16:06
  • $\begingroup$ Usually, text on probability, e.g. Kallenberg, Foundations of Modern Probability. But you can prove that lemma on the lines of Michael Greinecker's answer, using Beppo Levi's theorem. Consider the closure $V$ of the linear span of $1_{|A}$ with $A\in\Gamma$. Then, there are in $V$ all characteristic functions $1_{|B}$ with $B$ a set of finite measure that is a countable union of elements of $\Gamma$. And then, there are in $V$ all characteristic functions $1_{|C}$ with $C$ an intersection of a countable decreasing sequence $B_k$ of the preceding form (...) $\endgroup$
    – Pietro Majer
    Sep 3, 2012 at 17:28
  • $\begingroup$ .. and these sets $C$ are all the measurable subsets of finite measure, up to null measure sets. Thus $V$ has the characteristic functions of all sets of finite measure, therefore it is $L^p$ since it is closed. $\endgroup$
    – Pietro Majer
    Sep 3, 2012 at 17:30
  • $\begingroup$ sorry to ask once more. Why does C contain all sets except sets of measure zero? Moreover it is only clear that C contains disjoint countable union. Thank you? $\endgroup$
    – warsaga
    Sep 5, 2012 at 11:14
  • 1
    $\begingroup$ I mean: any measurable set $S$ of finite measure is of the form $C$ up to a perturbation of measure zero, that is, for any such $S$ there is a set $C$ such that $S\Delta C=0$ (that is $1_C=1_S$ a.e.) $\endgroup$
    – Pietro Majer
    Sep 6, 2012 at 20:29
1
$\begingroup$

All indicator functions of the algebra generated by rectangles are linear combinations of indicator functions of rectangles. Indicator functions of product measurable sets can be approximated by indicator functions of the algebra, which follows from the standard Caratheodory construction. Clearly, the same can be done with linear combinations of indicator functions.

Improve this answer
$\endgroup$
1
  • $\begingroup$ I know a result that the indicator functions of product measurable sets can be approximated by indicator functions of one variable. That is, for every $A\in F\otimes G$(product sigma algebra), there exist $F_k\in F$ and $G_k\in G$ such that $$ \sum_{k=1}^n 1_{F_k}1_{G_k} \to 1_{A}.$$ But I don not know how to prove it. Do you know a reference or give me some helps? By the way, can the convergence be monotonous? And can any measurable function $f$ defined on $X\times Y$ be approximated by linear combination of indicator functions of one variables monotonically? Many thanks! $\endgroup$
    – Longjie Xie
    Jun 12, 2014 at 13:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.